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Apologies to any formalist!

Here's the basic thought: $\mathbb{R}$ is a well-defined concept with unambiguous meaning in reality. Everyone can imagine an infinite series of digits (signifying the decimal representation of a real between $0$ and $1$ - let's please skip periods and other technicalities) - or perhaps let's just use binary sequences, whatever seems easier to imagine, so it seems rather clear what a real numbers is.

Similarly, consider the definition of a function: A relation that asigns an element of one set to exactly one element of another set - in this case an element of the reals to an element of the reals, or a bit more formal, an element of $\mathbb{R}^\mathbb{R}$, so this contains all possible mappings of reals onto other reals. This certainly includes all possible functions from subsets of $\mathbb{R}$ to $\mathbb{R}$. Now, whether such a function is a bijection is a clearly defined question as well, so that should, from this "realist" perspective, imply a defined value for CH and, more specifically, the cardinality of the reals.

Subsets and similar definitions appear similarly clearly defined.

Now, it seems clear to me why there's difficulties from any theory (in the strict formal mathematical, but also in any different sense) to establish truths about incountable sets - necessarily, anything humans can possibly state in any formal or informal language is countable, and as a result, there cannot be enough statements to ensure that for each real number there is a statement that is true for only that one specific real number. That's not even going into the unfathomably huge powerset of the reals where the relevant functions reside.

I know of research regarding this question in formal set theory, and I find it marvelous. But almost all of it is based on first-order logic, and most remotely relevant results happens in at least ZF, if not ZFC. It seems very obvious to me that it'll be hard at best to expect a definitive result on an uncountable set from essentially countable arguments - this makes the progress we have even more marvelous, of course, but it also seems hopeless to hold an ultimate answer.

There's certainly a lot of interesting results coming from this, but this question is not primarily about those results. Instead, it's two-fold:

a) Is there any (current, reasonable) results on the cardinality of the reals that is founded in the abstract (and hard to capture mathematically in a clearly consistent way) ideas above? This is explictly not restricted to research in first-order set theories, as explained above, and instead asks for results from different perspectives, if they exist.

b) Has there been any examination on how the results that come from strictly formal set theory compare to the above mentioned intuition? To explain: I've often seen, for instance, the claim that a certain forcing is "adding" sets - but adding sets and removing sets from the power set (i.e. the available functions) might easily be confused, and moreover the "base" (such as a countable model of ZFC) might already clearly not be relevant to "reality" as described above, so any derived results might not apply to "reality". Further, if someone looked for a specific real or function on reals as defined "in the intuitive sense" to be present in a certain model, would it always be there? Or perhaps, are there definable (in the broadest way possible) reals or functions of reals missing from some commonly considered models?

I completely understand that it's highly likely that these questions have no definitive answers, but I am curious about how mathematicians think about these questions.

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  • $\begingroup$ As for Platonic motivation for forcing, adding and removing reals (and really, the real numbers are just a small time player that got famous, it's a very small fish in the large ocean of sets), we study the limitations of $\sf ZFC$, and if we believe that $\sf ZFC$ is true in a Platonic sense, then forcing is used to study its limitations, what it cannot prove or disprove. For example, one might expect that $\sf ZFC$ proves the continuum hypothesis, but that's not the case; or that $\sf ZFC+CH$ proves $\lozenge$, but that's not the case either. $\endgroup$ – Asaf Karagila Oct 16 '14 at 2:14
  • $\begingroup$ @Bruce: I don't understand why you added the FOL tag. The question clearly states "This is explictly not restricted to research in first-order set theories, as explained above". $\endgroup$ – Asaf Karagila Oct 16 '14 at 2:16
  • $\begingroup$ What's interesting here is that there may be some intended model of ZFC (which is basically what the question asserts) - analogous to what's used for true arithmetic - but the difficulty is that we would probably have to accept some controversial axiom to prove it, since it seems that most intuitive properties are already in ZFC. The axiom of choice may very well be as valid as Peano arithmetic, but it's much harder to justify, from an intuitive standpoint. $\endgroup$ – Milo Brandt Oct 16 '14 at 2:17
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    $\begingroup$ @Meelo: The axiom of choice is insanely easy to justify from an intuitive point of view. You have a collection of non-empty sets, how can you not choose one from each? $\endgroup$ – Asaf Karagila Oct 16 '14 at 2:30
  • $\begingroup$ @Asaf It was, in any case, controversial for a while after its introduction, and it's not like there's no case against it - it does seem concerning that we can have proofs to the effect of, "This set exists, but cannot be constructed." In the purest sense, the statement is intuitive, but it implies very odd results, so it's not unreasonable to wonder if something's gone wrong in our intuition (esp. since our notion of a set & what ZFC has might not agree) I guess you're right - perhaps assumptions of the existence of inaccessible cardinals might be a better example of controversial axioms. $\endgroup$ – Milo Brandt Oct 16 '14 at 2:44
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Consider the following preprint on arXiv (arXiv: math/9209209v1 [math.LO]) by Garvin Melles:

"Natural Internal Forcing Schemata extending ZFC. A Crack in the Armor surrounding CH?"

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